Group #9720 — subgroups

Subgroups — #9720

Order 256.

211 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 211 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3170	C₂	1	1	yes	maximal, characteristic
2	#1986	C₂	1	2	yes	maximal
3	C₄×#106	C₂	1	1	yes	maximal, characteristic
4	#1503	C₂	1	2	yes	maximal
5	#1451	C₂	1	1	yes	maximal, characteristic
6	#533	C₂²	1	2	yes	—
7	#500	C₄	1	1	yes	characteristic, complemented
8	#494	C₄	1	1	yes	characteristic, complemented
9	#487	C₄	1	2	yes	complemented
10	#467	—	2	1	no	—
11	#467	—	2	1	no	—
12	C₄×Q₁₆	—	2	1	no	—
13	C₄×Q₁₆	—	2	1	no	—
14	#431	C₂²	1	1	yes	characteristic
15	C₂×#106	—	2	1	no	—
16	C₄×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄	C₂²	1	2	yes	—
19	#340	—	2	1	no	—
20	#339	—	2	4	no	—
21	#328	—	2	4	no	—
22	#322	—	2	1	no	—
23	#322	—	2	1	no	—
24	C₂×Q₁₆	—	2	1	no	—
25	C₂×Q₁₆	—	2	1	no	—
26	C₂×M₁₆	D₄	1	2	yes	—
27	C₂²×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₈	D₄	1	1	yes	characteristic
29	C₄⋊Q₈	C₂×C₄	1	2	yes	—
30	C₄⋊Q₈	C₂×C₄	1	2	yes	—
31	#125	—	2	2	no	—
32	C₂²⋊Q₈	—	2	2	no	—
33	C₂²⋊Q₈	—	2	2	no	—
34	C₄×Q₈	—	2	2	no	—
35	C₄×Q₈	—	2	2	no	—
36	#117	—	2	2	no	—
37	C₂×C₄⋊C₄	—	2	2	no	—
38	C₂×C₄²	—	2	2	no	—
39	C₂×C₄²	C₂³	1	1	yes	characteristic
40	#107	—	2	1	no	—
41	#107	—	2	1	no	—
42	#106	—	2	4	no	—
43	C₄⋊C₈	—	2	1	no	—
44	#103	—	2	2	no	—
45	#103	—	2	2	no	—
46	C₂²⋊C₈	D₄	1	2	yes	—
47	#97	—	2	1	no	—
48	C₄×C₈	—	2	1	no	—
49	C₄×C₈	—	2	1	no	—
50	C₄×C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈	D₄	1	1	yes	characteristic
53	C₄×C₈	D₄	1	1	yes	characteristic
54	#95	—	2	2	no	—
55	C₂×Q₈	—	2	2	no	—
56	C₂×Q₈	—	2	2	no	—
57	C₂²×C₄	—	4	2	no	—
58	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
61	Q₁₆	—	4	2	no	—
62	Q₁₆	—	4	2	no	—
63	M₁₆	—	2	4	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	4	1	no	—
68	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₄⋊C₄	—	2	4	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	4	2	no	—
78	C₄⋊C₄	—	4	2	no	—
79	C₄⋊C₄	—	4	2	no	—
80	C₂²⋊C₄	—	4	2	no	—
81	C₄²	—	4	2	no	—
82	C₄²	—	4	4	no	—
83	C₄²	C₂²×C₄	1	1	yes	characteristic
84	C₄²	C₂×D₄	1	1	yes	characteristic
85	C₄²	#41	1	1	yes	characteristic
86	C₄²	#41	1	1	yes	characteristic
87	C₂³	#127	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	8	2	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	—	4	4	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	8	2	no	—
98	C₂×C₄	—	8	2	no	—
99	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄	#121	1	1	yes	characteristic
103	C₂×C₄	#121	1	1	yes	characteristic
104	C₂×C₄	#124	1	1	yes	characteristic
105	C₂×C₄	#137	1	4	yes	—
106	C₈	—	2	1	no	—
107	C₈	—	2	1	no	—
108	C₈	—	2	1	no	—
109	C₈	—	4	2	no	—
110	C₈	—	4	2	no	—
111	C₈	#104	1	2	yes	—
112	C₂²	—	2	1	no	—
113	C₂²	—	2	1	no	—
114	C₂²	#390	1	1	yes	characteristic
115	C₂²	#497	1	2	yes	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	1	no	—
119	C₄	—	8	4	no	—
120	C₄	—	8	2	no	—
121	C₄	—	8	2	no	—
122	C₄	C₂×#104	1	1	yes	characteristic
123	C₄	#421	1	1	yes	characteristic
124	C₄	#441	1	2	yes	—
125	C₄	#470	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	#1971	1	1	yes	characteristic
128	C₂	#1976	1	1	yes	characteristic
129	C₂	#1980	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3170

C₂

yes

maximal, characteristic

#1986

C₂

yes

maximal

C₄×#106

C₂

yes

maximal, characteristic

#1503

C₂

yes

maximal

#1451

C₂

yes

maximal, characteristic

#533

C₂²

yes

—

#500

C₄

yes

characteristic, complemented

#494

C₄

yes

characteristic, complemented

#487

C₄

yes

complemented

#467

—

#467

—

C₄×Q₁₆

—

C₄×Q₁₆

—

#431

C₂²

yes

characteristic

C₂×#106

—

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

—

—

—

—

—

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

D₄

yes

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

C₂×C₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

—

—

—

C₄⋊C₈

—

#103

—

#103

—

C₂²⋊C₈

D₄

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

C₄²

yes

C₄²

yes

C₂³

yes

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

100

C₂×C₄

C₄×D₄

yes

characteristic

101

C₂×C₄

C₄×D₄

yes

characteristic

102

C₂×C₄

#121

yes

characteristic

103

C₂×C₄

#121

yes

characteristic

104

C₂×C₄

#124

yes

characteristic

105

C₂×C₄

#137

yes

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

#104

yes

—

112

C₂²

—

113

C₂²

—

114

C₂²

#390

yes

characteristic

115

C₂²

#497

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

C₂×#104

yes

characteristic

123

C₄

#421

yes

characteristic

124

C₄

#441

yes

—

125

C₄

#470

yes

—

126

C₂

—

127

C₂

#1971

yes

characteristic

128

C₂

#1976

yes

characteristic

129

C₂

#1980

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#390
prime radical	C₂³	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#121

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

C₂²

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
7. #500	C₄	semidirect
8. #494	C₄	semidirect
9. #487	C₄	semidirect

H ≅ G/N

type

7. #500

C₄

semidirect

8. #494

C₄

semidirect

9. #487

C₄

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#3170	2
#1986	2
C₄×#106	2
#1503	2
#1451	2

subgroup

index

Subgroups — #9720

Conjugacy classes of subgroups

Complements

Maximal subgroups